Control charts don't control
In reading your post you seem to be implying that the proper control chart will help you keep the true position at a minimum?
But, control charts only tell you if there is reason to believe the process has changed. They tell you if the true position for any one part is within the range expected from previous data.
I may be assuming, but I suspect that you like many others want to adjust the process based on the current reading or readings. I hope this is not the case, but if it is not I don't understand your comment about wanting a control chart that will maintain to the goal on the low end. The process will maintain to where it will maintain, regardless of the tool to monitor it. No control chart can change that, only process corrective action and improvement work.
As for the type to use, the golden rule is to capture all variation you want to call common cause within the subgroups and all the special causes between subgroups. This is called rational subgrouping. Since I think you are mostly interested in flagging abnormal true position, you would hide those to some extent by any grouping. It has to be an individual chart.